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Use the limit definition of the derivative to find a formula for f'(x) given f(x) = 3x^2 − 5

B. Use your result from part a to evaluate f'(− 1).

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Answer:

Explanation:

A. To find the formula for f'(x) using the limit definition of the derivative, we need to evaluate the following limit:

f'(x) = lim(h->0) [f(x + h) - f(x)] / h

Substituting f(x) = 3x^2 - 5, we get:

f'(x) = lim(h->0) [3(x + h)^2 - 5 - (3x^2 - 5)] / h

Expanding the square, simplifying, and canceling out the constant terms, we get:

f'(x) = lim(h->0) [6xh + 3h^2] / h

Canceling out the h's, we get:

f'(x) = lim(h->0) (6x + 3h)

Taking the limit as h approaches 0, we get:

f'(x) = 6x

Therefore, the formula for f'(x) is:

f'(x) = 6x

B. To evaluate f'(-1), we simply substitute x = -1 into the formula we found in part A:

f'(-1) = 6(-1) = -6

Therefore, f'(-1) = -6.

User Szymonm
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